Infinitive verb form

[Natalia]

I've come across two ways in which idioms are written (including infinitive verbs). For example:

1) to not see eye to eye with sb - to not agree with someone
2) not to see eye to eye with sb - to not agree with someone

Can you tell me which form is more correct?

[Daniel]

Neither is more "correct".

Generally the "to not" form sounds more natural, while prescriptivists tend to prefer the "not to" form to avoid the so-called 'split infinitive' to make English look more like Latin.

Both are used frequently, and preferences vary somewhat based on the specific construction. You could search a corpus (or just Google) to get a sense of frequencies for a particular usage, but either should be fine in almost all cases.

[Natalia]

Yes, I agree that 'to not' sounds more natural.
However, if I wrote down the idiom 'not to see eye to eye with sb' on the blackboard, should I also change the form in definition from 'to not agree with someone' to 'not to agree with someone' (for the sake of consistency)?
I mean:

1) not to see eye to eye with sb - not to agree with someone

Or it is fine to say:

2) not to see eye to eye with sb - to not agree with someone

[Daniel]

They're equivalent, so it's your choice.